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Sunday, December 27, 2015

Long before I became a parent, in my teaching (of community college students), a number of them told me how bad they were at math even though their mom or dad taught it. I figured the parents pushed too much or something. (Blame the parents much, do we?) I ‘knew’ I wouldn’t do that.

Well, I don’t think I pushed. But my son hates math, and is consequently way behind his peers. (He unschooled for years and there was no ‘behind’. But he chose to go to a regular middle school this year, where the other kids have mostly had the standard schooling.)
So when two people I respect got into a meaty conversation about this, my antennae popped up. They’ve allowed me to share this conversation, which occurred in a closed group on Facebook called 1001 Math Circles. (Ask to join if you’d like - group description: A place to share and discuss your #mathcircles plus learn more about the Natural Math principles! Run by Shelley Nash and Maria Droujkova of NaturalMath.com.)

Lhianna: Hi. I'm a homeschool mom of daughters 7 and 13. I absolutely love math and creative problem-solving and my oldest daughter hates it. My failure to transfer my love of math to her drove me to find better ways of teaching and sharing the beauty and excitement that I see. I found out about Math Circles and have done the summer training camp with Bob and Ellen Kaplan for several years now. I run Math Circles around Philadelphia as time and opportunity allow. I love getting inspired by all the great ideas of a wonderful math community like this one. Thanks for letting me join!

Maria: Lhianna, welcome! The Kaplans’ community is wonderful. Maybe we can have a live chat sometime about your circles?
When someone hates math, there is usually what I call a grief story. Even with homeschooling, our children can get enough grief "second-hand" from us, or from the society... When I ask people who hate math what happened to them, they usually do know, and tell their stories. Do you know what happened to your 13-year-old?
And what does your 7 year-old like to do? It's such interesting age for girls!

Lhianna: My 7 year-old loves logic problems. (The island of knights and knaves kind. I have a special fondness for all of Raymond Smullyan's books!) She likes unit origami (especially the sonobe units). And she seems fascinated by anything to do with parity. Also building with geometric shapes of all kinds.

I think my 13 year-old has a deep fear of getting things wrong in any subject and in general in life. In other subjects she finds ways around it. But it is especially devastating for mathematical exploration. You really have to try many different avenues and be able to look at your failures and analyze them to arrive at a solution in math. Math is about exploring what is unknown to you and she can't stand that. She prefers the familiar.

It has been an interesting journey for me. I started thinking how lucky she is to get an exploratory background in math. I then realized my own shortcomings that, while I loved to explore math, I hadn't been able to communicate that idea to my child. Which led me on a wonderful journey of discovering Math Circles and many more amazing people and sources full of creative ideas about learning math.

But as my daughter continued to hate it (and trying to do math with other people too, not just me), I also learned that math is not for everyone like I originally thought. It's ok now that she doesn't like math! That is a homeschooling journey to learn and accept this. (When she does do some math she is perfectly able to learn and understand the concepts. She just has zero interest and will not voluntarily spend any time on math study).

I am currently dragging her through "The Art of Problem Solving" book series so she can have enough math to go on to higher education. (And it's a pretty decent series for a textbook!) I am very much an amateur. I am constantly learning and open to new ideas. Any suggestions would be greatly helpful.

Maria: Lhianna, thank you for sharing. Yes, I am with you - love of math for its own sake isn't for everyone (just like any other area); but I do feel that everyone can feel good doing some math-rich activities in their own ways. I see a pattern in your interaction with math and with your 13 year-old. Do most of your math activities center on problem-solving?

In contrast, have you ever tried math activities that don't involve problems, solutions, answers, or unknowns? There are activities where you: (1) only work with what you know, and (2) don't seek any answers or solutions. When I say that now, can you picture 4-5 examples of activities that I am talking about?

Lhianna: Not off the top of my head. What kinds of activities are you thinking about?

Maria: Logic is so lovely! Smullyan's books made a difference for many people. Camp Logic, which we published this year, is one of our most popular books, too. I just sent three big boxes of it to groups. Next year, "Bright, Brave, Open Minds" will be out, by Julia Brodsky - there are very lovely logic activities in there, too. Here are a few things to try from that book:

Lhianna: I see my 13 year-old use math in other activities (she really likes to cook and make up her own recipes which involves experimentation and therefore doubling and tripling many measurements as well as analyzing the ratios of one ingredient to another). Is this what you are talking about?
Or math games? She likes to play SET.

Maria: Lhianna, so the goal is to find math-rich activities that: (1) are not problem-solving, and (2) center on what you already know, and yet (3) are open and can be made uniquely yours. Let’s see if we can find a fresh angle on what your daughter can try…

Storytelling. You tell what you know; you make the story interesting, fun, pretty, and may invent details, but you know your story (and math therein). Vi Hart videos are like that. Or storybooks like The Cat in Numberland.

Illustrations. Take something you know. Illustrate it with a picture, comic, video, toys, interpretive dance smile emoticon Basically, represent it by some medium you like. A lot of math comics are illustrations of math jokes, for example.

Programming. Take a formula or pattern you know and use, and make your computer (spreadsheet, solver, etc.) do it for you.

Scavenger hunt. Find some math idea you know (e.g. ratio) in what you like (e.g. Star Wars, your favorite park, or your room). Or find a lot of math ideas in one book, movie, room... Make a curated collection. There are a lot of those online.
Have you tried that sort of approach? How did it go?

SET is a very good game too. To use this as an example of doing what you like and know... We do this activity where we make our own set of SET cards from scratch, using our own shapes and themes. On the one hand, it's something you know. On the other, the amount of delicious a-ha moments you have along the way is just incredible!

Lhianna: Great idea! Thanks.
And thanks for the advice. I will start looking for activities and examples that follow along the lines of familiar but open. I appreciate the new perspective.

Maria: I would love to hear what else you find, because you have such a thoughtful approach to the whole thing! Moving the focus to, "Love SET, like Vi Hart videos, like Tangram puzzles..." (from, "hate math").

Do you have a kid who hates math? Do any of these ideas sound like something you might want to try out with them?

Wednesday, December 23, 2015

He determined the value of pi very precisely, by starting with a hexagon inscribed in a circle, then a 12-sided polygon, then he kept doubling the number of sides until he got to a 96-gon. A procedure like this is called the 'method of exhaustion', and it looks a lot like what we do nowadays with limits.

I am embarrassed to admit that I couldn't figure out how he did it. (I think I was focusing on area, and that might be harder.) I just found a great video by David Chandler (whose youtube channel is Math Without Borders).

Here's a summary:
Start with a hexagon inscribed in a circle of radius 1 (giving diameter of 2). The perimeter of the hexagon will be 6. This gives a lower bound on pi, which is the ratio of circumference to diameter. We know the circumferences is bigger than this perimeter of 6, so pi is bigger than 6/2 = 3.

If you cut one of the triangles that made the hexagon into two, you get a radius that crosses a side of the hexagon at right angles. You can use the Pythagorean Theorem (twice) to find the new side length. Repeat 3 times and you're at the 96-gon. Archimedes had none of our technology, and little or none of our algebraic symbolism, so the calculations were much harder for him. We can do all this on a spreadsheet, and up comes pi (if you have a column for the perimeter over the diameter). So satisfying!

Dan MacKinnon wrote a lovely review of a book I hadn't heard of before, at his blog, Math Recreation. Here's the beginning of it...

In The Puzzle Universe: A History of Mathematics* in 315 Puzzles (TPU), Ivan Moscovich stretches the concept of puzzles to encompass almost anything that combines curiosity and playfulness (playthinks
is his preferred term for this more general category of puzzling
items). No surprise - these playful curiosities are inherently
mathematical. In an informal and accessible way, Moscovich details the
development of these puzzles, revealing their surprising family
resemblances and the deep mathematics behind their playful exterior. [read the rest at Dan's blog...]

_________*This link goes to bookfinder.com, which will point to other sites. It's the best way I know of to find the least expensive copy available. (My other links point to the sites that were cheapest at bookfinder on the day I wrote this.)